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By Hartmut Laue

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Fn ∈ H. Then ci = 0R for all i ∈ n. Exploiting our R-linear relation for the fi in two different ways we obtain, for arbitrary x, x′ ∈ X, the equations ci (x′ fi )(xfi ) = ci (x′ x)fi = 0R , i∈n i∈n ′ ci (x fn )(xfi ) = x′ fn i∈n ci (xfi ) = 0R , i∈n 35 ′ ′ ′ i∈n−1 ci (x fi −x fn )(xfi ) = 0R by subtraction. For arbitrary x ∈ X we conclude ′ ′ i∈n−1 ci x (fi − fn ) fi = 0RX , implying ci x (fi −fn ) = 0R for every i ∈ n, by the choice hence of n. For all i ∈ n we have ci = 0R , hence fi − fn = 0RX , i.

Tn ) in K(t1 , . . , tn ). From the definition of Ksym [t1 , . . , tn ] we obtain that Ksym (t1 , . . , tn ) is the set of all fixed points with respect to the set of automorphisms induced by the symmetric group Sn . 2. Let n ∈ N, K a field. Then there exists a monomorphism of Sn into the subgroup {α|α ∈ Aut K(t1 , . . , tn ), ∀q ∈ Ksym (t1 , . . , tn ) qα = q} of Aut K(t1 , . . , tn ). Later (see p. 73) we will see that this monomorphism is in fact an isomorphism, in other words, that every automorphism of K(t1 , .

Terminological inconsistencies of this kind often occur as a result of an attempted compromise between conceptual clearness and certain traditions of mathematical language. ,bn is not only closed but has the much stronger property of being an ideal of P (cf. 5). Clearly, in a commutative algebra there is no distinction between left and right ideals. 40 39 Any n-tuple (b1 , . . , bn ) ∈ B n with this property is called a zero of f , generalizing the definition given on p. 9. The fundamental topic of Algebraic Geometry is to study the set of common zeros in B n for a subset of P .

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